Method for measuring rheological property of drilling fluid by using curved pipe on site

ABSTRACT

A method for measuring a rheological property of a drilling fluid by using a curved pipe on site includes: step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking; step 2: calculating Rei according to fci; step 3: calculating an actual shear stress τwi and a shear rate γi of the drilling fluid in the on-site curved pipe according to the relationship constants and Rei; step 4: establishing a plurality of on-site models according to τwi and γi; step 5: determining an optimal on-site model according to correlations between τwi and predicted shear stresses of the plurality of on-site models; and step 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model. The method avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids.

CROSS REFERENCE TO THE RELATED APPLICATIONS

This application is the continuation application of International Application No. PCT/CN2020/138476, filed on Dec. 23, 2020, the entire contents of which are incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to the field of oil well construction and, in particular, to a method for measuring a rheological property of a drilling fluid by using a curved pipe on site.

BACKGROUND

The properties of the drilling fluid (mud) play a crucial role in optimizing drilling operations. Among them, the density and rheological properties of the drilling fluid play a significant role in the optimal management of wellbore pressure. Therefore, it is necessary to carry out accurate measurement of the density and rheological properties of the drilling fluid in narrow-window drilling operations, especially in advanced drilling techniques, including managed pressure drilling (MPD) and dual gradient drilling (DGD).

At present, the rheological properties of the drilling fluid are mainly measured by a rotation method and a pipe flow method. In the pipe flow method, the online rheological measurement device uses a straight pipe for measurement. However, due to its large size, it requires significant changes to the site space, resulting in greatly limited on-site applications and poor measurement accuracy.

SUMMARY

A technical problem to be solved by the present invention is to provide a method for measuring a rheological property of a drilling fluid by using a curved pipe on site, so as to improve the accuracy of the rheological measurement of the drilling fluid.

To solve the above technical problem, the present invention adopts the following technical solution: a method for measuring a rheological property of a drilling fluid by using a curved pipe on site, including the following steps:

step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking;

step 2: calculating a Reynolds number R_(ei) of the drilling fluid in an on-site curved pipe according to a friction coefficient f_(ci) of the drilling fluid in the on-site curved pipe;

step 3: calculating an actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe, where i denotes a number of times the drilling fluid flows through the on-site curved pipe, which is a positive integer not less than 2;

step 4: establishing a plurality of on-site models according to the actual shear stress τw_(i) and a shear rate γ of the drilling fluid;

step 5: determining an optimal on-site model according to correlations between the actual shear stress τw_(i) and predicted shear stresses of the plurality of on-site models; and

step 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.

The working principle and beneficial effects of the present invention are as follows: The present invention derives the relationship constants between the friction coefficients of the drilling fluid through offline checking and can derive the relationship constants for different types of drilling fluids. The present invention avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids. In addition, the present invention determines the optimal on-site model according to the correlations between the actual shear stress τw_(i) and the predicted shear stresses of the plurality of on-site models, so as to ensure the accuracy of on-site measurement of the drilling fluid.

The following improvement may be further made by the present invention based on the above technical solution.

Further, step 1 may include:

step 11: calculating a friction coefficient f_(ck) of the drilling fluid in an offline curved pipe and a friction coefficient f_(sk) of the drilling fluid in an offline straight pipe, where, k denotes a number of times the drilling fluid flows through an offline pipe, which is a positive integer not less than 2;

step 12: establishing a plurality of prediction models according to an actual friction coefficient ratio y_(k), where y_(k)=f_(ck)/f_(sk);

step 13: determining an optimal prediction model according to correlations between the actual friction coefficient ratio y_(k) and predicted friction coefficient ratios of the plurality of prediction models; and

step 14: deriving the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.

The beneficial effects of the above further solution are as follows. The present invention measures and calculates the actual friction coefficients of the drilling fluid in offline curved and straight pipes and establishes the plurality of prediction models. The present invention determines the optimal prediction model according to the correlations between the actual friction coefficient ratio y_(i) and the predicted friction coefficient ratios of the plurality of prediction models. The present invention ensures the accuracy of selecting the prediction model. The present invention uses the relationship constants between the friction coefficients of the selected prediction model for subsequent on-site measurement on the rheological property of the drilling fluid. The present invention analyzes the correlations through the plurality of prediction models to ensure the accuracy of the relationship constants between the friction coefficients of different types of drilling fluids. Therefore, the present invention avoids the inaccuracy caused when a single relationship constant between the friction coefficients is applied to the on-site measurement of the drilling fluid.

The following improvement may be further made by the present invention based on the above technical solution.

Further, in step 11, f_(ck) may be expressed by formula (1):

$\begin{matrix} {f_{ck} = {\frac{d_{{tc}\; 1}}{2\rho_{1}*v_{ck}^{2}}*\frac{\Delta\; P_{ck}}{\Delta L_{ck}}}} & (1) \end{matrix}$

where, d_(tc1) denotes an inner diameter of the offline curved pipe, and has a unit of m;

ρ₁ denotes a density of an offline drilling fluid, and has a unit of kg/m³;

v_(ck) denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s; and

ΔP_(ck)/ΔL_(ck) denotes a measured average pressure difference in the offline curved pipe, and has a unit of kPa/m; and ΔP_(ck) denotes a total pressure difference in a pipe section with a length of ΔL_(ck), and has a unit of kPa;

in step 1, f_(sk) may be expressed by formula (2):

$\begin{matrix} {f_{sk} = {\frac{d_{{ts}\; 1}}{2\rho_{1}*v_{sk}^{2}}*\frac{\Delta\; P_{sk}}{\Delta L_{sk}}}} & (2) \end{matrix}$

where, d_(ts1) denotes an inner diameter of the offline straight pipe, and has a unit of m;

ρ₁ denotes a density of the drilling fluid, and has a unit of kg/m³;

v_(sk) denotes a flow velocity of the drilling fluid at the k-th time in the offline straight pipe, and has a unit of m/s; and

ΔP_(sk)/ΔL_(sk) denotes a measured average pressure difference in the offline straight pipe, and has a unit of kPa/m; and ΔP_(sk) denotes a total pressure difference in a pipe section with a length of ΔL_(sk), and has a unit of kPa.

The beneficial effects of the above further solution are as follows. The present invention measures the average pressure differences of the straight pipe and the curved pipe and calculates the friction coefficient f_(ck) of the drilling fluid in the curved pipe and the friction coefficient f_(sk) thereof in the straight pipe, so as to ensure the calculation accuracy of the friction coefficients.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

there may be at least three prediction models, namely:

a first prediction model:

ŷ _(1k) =a*D _(nk) ^(b) +C  (3)

a second prediction model:

$\begin{matrix} {{\hat{y}}_{2k} = {1 + \frac{a*D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$

a third prediction model:

ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5)

where

ŷ_(1k) denotes a predicted friction coefficient of the first prediction model;

ŷ_(2k) denotes a predicted friction coefficient of the second prediction model;

ŷ_(3k) denotes a predicted friction coefficient of the third prediction model; and

a, b and c denote the relationship constants between the friction coefficients of the drilling fluid, respectively;

where, D_(nk) denotes a Dean number of the drilling fluid at the k-th time in the offline curved pipe, and is expressed by formula (6):

$\begin{matrix} {D_{nk} = {\frac{\rho_{1}*v_{ck}*d_{{tc}\; 1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}\; 1}}{D_{c1}}}}} & (6) \end{matrix}$

where

μ₁ denotes a viscosity of an offline drilling fluid, and has a unit of Pa·s; and

v_(ck) denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s.

The advantages of the above further solution are as follows. The present invention designs a plurality of prediction models involving the Dean number, which ensures the calculation accuracy.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

step 13 may specifically include:

expressing a correlation R₁₁ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the first prediction model by formula (7):

$\begin{matrix} {R_{11}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{1k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (7) \end{matrix}$

expressing a correlation R₁₂ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model by formula (8):

$\begin{matrix} {R_{12}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{2k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (8) \end{matrix}$

expressing a correlation R₁₃ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model by formula (9):

$\begin{matrix} {R_{13}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{3k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (9) \end{matrix}$

comparing R₁₁ ², R₁₂ ² and R₁₃ ² in terms of magnitude, and selecting a prediction model with a maximum correlation as an optimal prediction model;

where

m denotes a number of samples;

y_(k) denotes the actual friction coefficient ratio; and

y denotes an average actual friction coefficient ratio.

The beneficial effects of the above further solution are as follows: The present invention selects the final model for offline calibration through the correlations between the actual friction coefficient ratio y_(i) and the predicted friction coefficient ratios of the prediction models, which ensures the accuracy of selecting the offline model.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 2, f_(ci) may be expressed by formula (10):

$\begin{matrix} {f_{ci} = {\frac{d_{{tc}\; 2}}{2\rho_{2}*v_{ci}^{2}}*\frac{\Delta\; P_{ci}}{\Delta L_{ci}}}} & (10) \end{matrix}$

where, d_(tc2) denotes an inner diameter of the on-site curved pipe, and has a unit of m;

ρ₂ denotes a density of an on-site drilling fluid, and has a unit of kg/m³;

v_(ci) denotes a flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ΔP_(ci)/ΔL_(ci) denotes a measured average pressure difference in the on-site curved pipe, and has a unit of kPa/m; and ΔP_(ci) denotes a total pressure difference in a pipe section with a length of ΔL_(ci), and has a unit of kPa.

The beneficial effects of the above further solution are as follows. The present invention can accurately calculate the friction coefficient f_(ci) of the drilling fluid in the on-site curved pipe.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 2, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe may be calculated as follows:

when an offline model is the first prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (12):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}\left( {{a*\left( {R_{ei}*\sqrt{\frac{d_{{tc}\; 2}}{D_{c2}}}} \right)^{b}} + c} \right)}} & (12) \end{matrix}$

when the offline model is the second prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (13):

$\begin{matrix} {{f_{ci} = {\frac{16}{R_{ei}}*\left\lbrack {1 + \frac{a*\left( {R_{ei}*\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)^{b}}{{70} + \left( {R_{ei}*\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}} \right\rbrack}},} & (13) \end{matrix}$

and

when the offline model is the third prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (14):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}*{\left\lbrack {1 + {a*\left( {\log_{10}\left( {R_{ei}*\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)} \right)^{b}}} \right\rbrack.}}} & (14) \end{matrix}$

The beneficial effects of the above further solution are as follows. The present invention calculates the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe and adopts different calculation methods according to different offline models, thereby ensuring the accuracy of the calculation of the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 3, the actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe may be expressed by formula (15):

$\begin{matrix} {\tau_{wi} = \frac{8\rho_{2}*v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$

where

v_(ci) denotes the flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ρ₂ denotes the density of the on-site drilling fluid, and has a unit of kg/m³;

the shear rate γ_(i) of the drilling fluid is expressed by formula (16):

$\begin{matrix} {\gamma_{i} = {\frac{8*v_{ci}}{d_{{tc}\; 2}}*\frac{{3*N_{i}} + 1}{4*N_{i}}}} & (16) \end{matrix}$

where, N is expressed by formula (17):

$\begin{matrix} {N_{i} = \frac{d\left( {\ln\tau_{w_{i}}} \right)}{d\left( {\ln\frac{8*v_{ci}}{d_{{tc}\; 2}}} \right)}} & (17) \end{matrix}$

The beneficial effects of the above further solution are as follows. The present invention ensures the calculation accuracy.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

there may be at least three on-site models, namely:

a first on-site model:

τŵ _(1i) =YP−PV*γ _(i)  (18)

a second on-site model:

τŵ _(2i) =K*γ _(i) ^(n)  (19)

a third on-site model:

τŵ _(2i)=τ₀ +K*γ _(i) ^(n)  (20)

where

YP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa;

PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s;

n denotes a fluidity index of the on-site drilling fluid, and is dimensionless;

K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; and

τ₀ denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.

The beneficial effects of the above further solution are as follows. The present invention selects the rheological parameters through three different on-site models to ensure the optimal on-site models available for different drilling fluids.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

step 5 may specifically include:

expressing a correlation R₂₁ ² between the actual shear stress τw_(i) and a predicted shear stress of the first on-site model by formula (21):

$\begin{matrix} {R_{21}^{2} = {1 - \frac{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - {\tau\;{\hat{w}}_{1i}}} \right)^{2}}{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (21) \end{matrix}$

expressing a correlation R₂₂ ² between the actual shear stress τw_(i) and a predicted shear stress of the second on-site model by formula (22):

$\begin{matrix} {R_{22}^{2} = {1 - \frac{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - {\tau\;{\hat{w}}_{2i}}} \right)^{2}}{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (22) \end{matrix}$

expressing a correlation R₂₃ ² between the actual shear stress τw_(i) and a predicted shear stress of the third on-site model by formula (23):

$\begin{matrix} {R_{23}^{2} = {1 - \frac{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - {\tau\;{\hat{w}}_{3i}}} \right)^{2}}{\overset{m}{\sum\limits_{i = 1}}\left( {{\tau w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (23) \end{matrix}$

comparing R₂₁ ², R₂₂ ² and R₂₃ ² in terms of magnitude, and selecting an on-site model with a maximum correlation as a final model;

where

m denotes a number of samples;

τw_(i) denotes the actual shear stress; and

τw denotes an average actual shear stress.

The beneficial effects of the above further solution are as follows: The present invention selects the final model for on-site measurement through the correlations between the actual shear stress τw_(i) and the predicted friction coefficient ratios of the on-site models, ensuring the accuracy of selecting the on-site model.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart of an on-site measurement method according to a first embodiment of the present invention; and

FIG. 2 is a flowchart of an offline checking method according to the first embodiment of the present invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Principles and features of the present invention are described below with reference to the drawings. The described embodiments are only used to explain the present invention, rather than to limit the scope of the present invention.

FIG. 1 is a flowchart of an on-site measurement method according to a first embodiment of the present invention.

A method for measuring a rheological property of a drilling fluid by using a curved pipe on site includes the following steps:

Step 1: Derive relationship constants between friction coefficients of a drilling fluid through offline checking.

Step 2: Calculate a Reynolds number R_(ei) of the drilling fluid in an on-site curved pipe according to a friction coefficient f_(ci) of the drilling fluid in the curved pipe.

Step 3: Calculate an actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe, where i denotes a number of times the drilling fluid flows through the on-site curved pipe, which is a positive integer not less than 2.

Step 4: Establish a plurality of on-site models according to the actual shear stress τw_(i) and a shear rate γ of the drilling fluid.

Step 5: Determine an optimal on-site model according to correlations between the actual shear stress τw_(i) and predicted shear stresses of the plurality of on-site models.

Step 6: Perform on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.

The working principle and beneficial effects of the embodiment of the present invention are as follows. The present invention derives the relationship constants between the friction coefficients of the drilling fluid through offline checking and can derive the relationship constants for different types of drilling fluids. The present invention avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids. In addition, the present invention determines the optimal on-site model according to the correlations between the actual shear stress τw_(i) and the predicted shear stresses of the plurality of on-site models, so as to ensure the accuracy of on-site measurement of the drilling fluid.

In this embodiment, when the on-site measurement on the rheological property of the drilling fluid is carried out by the optimal on-site model, if the friction of the curved pipe changes, Step 7 is performed. That is, according to the friction of the curved pipe, the entire method starting from Step 1 is repeated outside a fixed operation time. The fixed operation time refers to a normal operation time of the on-site measurement equipment.

FIG. 2 is a flowchart of an offline checking method according to the embodiment of the present invention.

Step 1 includes:

Step 11: Calculate a friction coefficient f_(ck) of the drilling fluid in an offline curved pipe and a friction coefficient f_(sk) of the drilling fluid in an offline straight pipe, where, k denotes a number of times the drilling fluid flows through an offline pipe, which is a positive integer not less than 2.

Step 12: Establish a plurality of prediction models according to an actual friction coefficient ratio y_(k), where, y_(k)=f_(ck)/f_(sk).

Step 13: Determine an optimal prediction model according to correlations between the actual friction coefficient ratio y_(k) and predicted friction coefficient ratios of the plurality of prediction models.

Step 14: Derive the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.

The present invention measures and calculates the actual friction coefficients of the drilling fluid in offline curved and straight pipes and establishes the plurality of prediction models. The present invention determines the optimal prediction model according to the correlations between the actual friction coefficient ratio y_(i) and the predicted friction coefficient ratios of the plurality of prediction models. The present invention ensures the accuracy of selecting the prediction model. The present invention uses the relationship constants between the friction coefficients of the selected prediction model for subsequent on-site measurement on the rheological property of the drilling fluid. The present invention analyzes the correlations through the plurality of prediction models to ensure the accuracy of the relationship constants between the friction coefficients of different types of drilling fluids. Therefore, the present invention avoids the inaccuracy caused when a single relationship constant between the friction coefficients is applied to the on-site measurement of the drilling fluid.

Specifically, in step 11, f_(ck) is expressed by formula (1):

$\begin{matrix} {f_{ck} = {\frac{d_{{tc}\; 1}}{2\rho_{1}*v_{ck}^{2}}*\frac{\Delta\; P_{ck}}{\Delta L_{ck}}}} & (1) \end{matrix}$

where, d_(tc1) denotes an inner diameter of the offline curved pipe, and has a unit of m;

ρ₁ denotes a density of an offline drilling fluid, and has a unit of kg/m³;

v_(ck) denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s; and

ΔP_(ck)/ΔL_(ck) denotes a measured average pressure difference in the offline curved pipe, and has a unit of kPa/m; and ΔP_(ck) denotes a total pressure difference in a pipe section with a length of ΔL_(ck), and has a unit of kPa;

in step 1, f_(sk) is expressed by formula (2):

$\begin{matrix} {f_{ck} = {\frac{d_{{ts}\; 1}}{2\rho_{1}*v_{sk}^{2}}*\frac{\Delta\; P_{sk}}{\Delta L_{sk}}}} & (2) \end{matrix}$

where, d_(ts1) denotes an inner diameter of the offline straight pipe, and has a unit of m;

ρ₁ denotes a density of the drilling fluid, and has a unit of kg/m³;

v_(sk) denotes a flow velocity of the drilling fluid at the k-th time in the offline straight pipe, and has a unit of m/s; and

ΔP_(sk)/ΔL_(sk) denotes a measured average pressure difference in the offline straight pipe, and has a unit of kPa/m; and ΔP_(sk) denotes a total pressure difference in a pipe section with a length of ΔL_(sk), and has a unit of kPa.

where,

$\begin{matrix} {d_{{tc}\; 1} = \sqrt{\frac{4*V_{1}}{\pi*{le}n_{1}}}} & (24) \end{matrix}$

where, V₁ denotes a total volume of the offline curved pipe, and has a unit of m³; and

len₁ denotes an length of the offline curved pipe, and has a unit of m.

In this embodiment, there are at least three prediction models, namely:

a first prediction model:

ŷ _(1k) =a*D _(nk) ^(b) +c  (3)

a second prediction model:

$\begin{matrix} {{\hat{y}}_{2k} = {1 + \frac{a*D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$

a third prediction model:

ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5)

where

ŷ_(1k) denotes a predicted friction coefficient of the first prediction model;

ŷ_(2k) denotes a predicted friction coefficient of the second prediction model;

ŷ_(3k) denotes a predicted friction coefficient of the third prediction model; and

a, b and c denote the relationship constants between the friction coefficients of the drilling fluid, respectively;

where, D_(nk) denotes a Dean number of the drilling fluid at the k-th time in the offline curved pipe, and is expressed by formula (6):

$\begin{matrix} {D_{nk} = {\frac{\rho_{1}*v_{ck}*d_{{tc}\; 1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}\; 1}}{D_{c\; 1}}}}} & (6) \end{matrix}$

where

μ₁ denotes a viscosity of an offline drilling fluid, and has a unit of Pa·s; and

v_(ck) denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s.

Step 13 specifically includes:

Express a correlation R₁₁ ² between the actual friction coefficient ratio yk and a predicted friction coefficient ratio of the first prediction model by formula (7):

$\begin{matrix} {R_{11}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{1k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (7) \end{matrix}$

express a correlation R122 between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model by formula (8):

$\begin{matrix} {R_{12}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{2k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (8) \end{matrix}$

express a correlation R132 between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model by formula (9):

$\begin{matrix} {R_{13}^{2} = {1 - \frac{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \hat{y_{3k}}} \right)^{2}}{\overset{m}{\sum\limits_{k = 1}}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (9) \end{matrix}$

compare R₁₁ ², R₁₂ ² and R₁₃ ² in terms of magnitude, and select a prediction model with a maximum correlation as an optimal prediction model;

where

m denotes a number of samples;

y_(k) denotes the actual friction coefficient ratio; and

y denotes an average actual friction coefficient ratio.

In this embodiment, in step 2, f_(ci) is expressed by formula (10):

$\begin{matrix} {f_{ci} = {\frac{d_{tc2}}{2\rho_{2}*v_{ci}^{2}}*\frac{\Delta\; P_{ci}}{\Delta L_{ci}}}} & (10) \end{matrix}$

where, d_(tc2) denotes an inner diameter of the on-site curved pipe, and has a unit of m;

ρ₂ denotes a density of an on-site drilling fluid, and has a unit of kg/m³;

v_(ci) denotes a flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ΔP_(ci)/ΔL_(ci) denotes a measured average pressure difference in the on-site curved pipe, and has a unit of kPa/m; and ΔP_(ci) denotes a total pressure difference in a pipe section with a length of ΔL_(ci), and has a unit of kPa; and

d_(tc2) is expressed by formula (11):

$\begin{matrix} {d_{tc2} = \sqrt{\frac{4*V_{2}}{\pi*{len}_{2}}}} & (11) \end{matrix}$

where, V₂ denotes a total volume of the on-site curved pipe, and has a unit of m³; and

len₂ denotes a length of the on-site curved pipe, and has a unit of m;

Specifically, in step 2, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe is calculated as follows:

when an offline model is the first prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (12):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}\left( {{a*\left( {R_{ei}*\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)^{b}} + c} \right)}} & (12) \end{matrix}$

when the offline model is the second prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (13):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}*\left\lbrack {1 + \frac{{a^{*}\left( {R_{ei}*\sqrt{\frac{d_{{tc}2}}{D_{c2}}}} \right)}^{b}}{{70} + \left( {R_{ei}*\sqrt{\frac{d_{{tc}2}}{D_{c2}}}} \right)}} \right\rbrack}} & (13) \end{matrix}$

when the offline model is the third prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (14):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}*{\left\lbrack {1 + {a^{*}\left( {\log_{10}\left( {R_{ei}*\sqrt{\frac{d_{{tc}2}}{D_{c2}}}} \right)} \right)}^{b}} \right\rbrack.}}} & (14) \end{matrix}$

Specifically, in step 3, the actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe is expressed by formula (15):

$\begin{matrix} {\tau_{wi} = \frac{{8\rho_{2}} \star v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$

where

v_(ci) denotes the flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ρ₂ denotes the density of the on-site drilling fluid, and has a unit of kg/m³;

the shear rate γ_(i) of the drilling fluid is expressed by formula (16):

$\begin{matrix} {\gamma_{i} = {\frac{8^{*}v_{ci}}{d_{tc2}}*\frac{{3^{*}N_{i}} + 1}{4^{*}N_{i}}}} & (16) \end{matrix}$

where, N is expressed by formula (17):

$\begin{matrix} {N_{i} = \frac{d\left( {\ln\tau_{w_{i}}} \right)}{d\left( {\ln\frac{8^{*}v_{ci}}{d_{tc2}}} \right)}} & (17) \end{matrix}$

In this embodiment, there are at least three on-site model, namely:

a first on-site model:

{circumflex over (τ)}w _(1i) =YP+PV*γ _(i)  (18)

a second on-site model:

{circumflex over (τ)}w _(2i) =K*γ _(i) ^(n)  (19)

a third on-site model:

{circumflex over (τ)}w _(2i)=τ₀ K*γ _(i) ^(n)  (20)

where

YP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa;

PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s;

n denotes a fluidity index of the on-site drilling fluid, and is dimensionless;

K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; and

τ₀ denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.

Specifically, step 5 includes:

express a correlation R₂₁ ² between the actual shear stress τw_(i) and a predicted shear stress of the first on-site model by formula (21):

$\begin{matrix} {R_{21}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - {\tau{\overset{\hat{}}{w}}_{1i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - \overset{\_}{\tau w}} \right)^{2}}}} & (21) \end{matrix}$

express a correlation R₂₂ ² between the actual shear stress τw_(i) and a predicted shear stress of the second on-site model by formula (22):

$\begin{matrix} {R_{22}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - {\tau{\overset{\hat{}}{w}}_{2i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - \overset{\_}{\tau w}} \right)^{2}}}} & (22) \end{matrix}$

express a correlation R₂₃ ² between the actual shear stress τw_(i) and a predicted shear stress of the third on-site model by formula (23):

$\begin{matrix} {R_{23}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - {\tau{\overset{\hat{}}{w}}_{3i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w_{i}} - \overset{\_}{\tau w}} \right)^{2}}}} & (23) \end{matrix}$

compare R₂₁ ², R₂₂ ² and R₂₃ ² in terms of magnitude, and selecting an on-site model with a maximum correlation as a final model;

where

m denotes a number of samples;

τw_(i) denotes the actual shear stress; and

τw denotes an average actual shear stress.

The application of the first embodiment of offline checking of the present invention is described below.

In this embodiment, the offline curved pipe is a helical pipe, which has a total volume V₁=1.04 l and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_(ts1)=0.01056 m; and a first offline drilling fluid has a density ρ₁=1,003 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_(tc1)=0.01051 m:

$\begin{matrix} {d_{{tc}1} = \sqrt{\frac{4^{*}V_{1}}{\pi^{*}{len}_{1}}}} & (4) \end{matrix}$

In this embodiment, the first offline drilling fluid flows through the pipe for k=24 times.

A flow velocity v_(sk) of the first offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_(ck) thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_(ck)/ΔL_(ck) of the offline curved pipe and an average pressure difference ΔP_(sk)/ΔL_(sk) of the offline straight pipe are also shown in the table below.

The first offline drilling fluid flows through for 24 times, and its friction coefficient f_(ck) in the offline curved pipe and friction coefficient f_(sk) in the offline straight pipe are calculated by formulas (1) and (2) and are shown in the table below.

$\begin{matrix} {f_{ck} = {\frac{d_{{tc}1}}{{2\rho_{1}} \star v_{ck}^{2}}*\frac{\Delta P_{ck}}{\Delta L_{ck}}}} & (1) \end{matrix}$

$\begin{matrix} {f_{sk} = {\frac{d_{{ts}1}}{{2\rho_{1}} \star v_{sk}^{2}}*\frac{\Delta P_{sk}}{\Delta L_{sk}}}} & (2) \end{matrix}$

The corresponding actual friction coefficient ratios y_(k) are shown in the table below, where y_(k)=f_(ck)/f_(sk).

In this embodiment, an average viscosity of the first offline drilling fluid is μ₁=0.00711 Pa·s.

The Dean numbers D_(nk) of the first offline drilling fluid flowing through the curved pipe for 24 times are calculated according to formula (6) and are shown in the table below.

$\begin{matrix} {D_{nk} = {\frac{\rho_{1} \star v_{ck} \star d_{{tc}1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}1}}{D_{c1}}}}} & (6) \end{matrix}$

Number ΔP_(ck)/ ΔP_(sk)/ of times ΔL_(ck) ΔL_(sk) v_(sk) _(vck) f_(sk) f_(ck) y_(k) D_(nk) 1 0.576 0.590 0.169 0.170 0.106 0.106 1.001  29.557 2 0.651 0.727 0.235 0.237 0.062 0.068 1.091  50.588 3 0.674 0.777 0.264 0.266 0.051 0.058 1.127  61.461 4 0.691 0.830 0.274 0.276 0.049 0.057 1.175  64.702 5 0.718 0.873 0.304 0.307 0.041 0.049 1.187  76.722 6 0.748 0.927 0.327 0.330 0.037 0.045 1.211  85.333 7 0.777 0.979 0.348 0.351 0.034 0.042 1.231  92.785 8 0.807 1.023 0.374 0.377 0.030 0.038 1.240 103.122 9 0.827 1.064 0.397 0.401 0.028 0.035 1.257 113.686 10 0.909 1.169 0.440 0.444 0.025 0.031 1.257 126.935 11 0.936 1.242 0.482 0.486 0.021 0.028 1.297 147.922 12 0.973 1.295 0.503 0.508 0.020 0.026 1.301 155.213 13 0.999 1.350 0.543 0.548 0.018 0.024 1.321 176.185 14 1.031 1.408 0.571 0.576 0.017 0.022 1.334 188.652 15 1.071 1.468 0.591 0.597 0.016 0.022 1.340 194.715 16 1.111 1.525 0.616 0.622 0.015 0.021 1.341 203.773 17 1.111 1.589 0.631 0.637 0.015 0.021 1.398 213.799 18 1.142 1.627 0.665 0.671 0.014 0.019 1.393 230.859 19 1.178 1.694 0.696 0.703 0.013 0.018 1.405 245.566 20 1.196 1.785 0.723 0.729 0.012 0.018 1.459 260.502 21 1.243 1.845 0.769 0.776 0.011 0.016 1.450 283.448 22 1.262 1.904 0.778 0.785 0.011 0.016 1.475 286.099 23 1.294 1.977 0.809 0.817 0.010 0.016 1.494 301.892 24 1.360 2.089 0.879 0.887 0.009 0.014 1.502 338.859

Three prediction models are fitted by the actual friction coefficient ratios yk and the Dean numbers D_(nk) as follows:

a first prediction model:

ŷ _(1k) =a*D _(nk) ^(b) +c  (3)

where, a=0.035966, b=0.5, and c=0.855298.

a second prediction model:

$\begin{matrix} {{\overset{\hat{}}{y}}_{2k} = {1 + \frac{a^{*}D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$

where, a=0.052896, and b=1.421332.

a third prediction model:

ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5)

where, a=0.016495, and b=3.709336.

The predicted friction coefficients of the third prediction models are shown in the table below.

Number of times ŷ_(1k) ŷ_(2k) ŷ_(3k) 1 1.051 1.065 1.069 2 1.111 1.116 1.119 3 1.137 1.140 1.143 4 1.145 1.147 1.149 5 1.170 1.172 1.173 6 1.188 1.189 1.189 7 1.202 1.203 1.203 8 1.221 1.222 1.221 9 1.239 1.241 1.239 10 1.261 1.262 1.260 11 1.293 1.295 1.292 12 1.303 1.305 1.303 13 1.333 1.335 1.332 14 1.349 1.351 1.348 15 1.357 1.359 1.356 16 1.369 1.370 1.368 17 1.381 1.382 1.380 18 1.402 1.402 1.401 19 1.419 1.418 1.418 20 1.436 1.434 1.435 21 1.461 1.458 1.460 22 1.464 1.461 1.463 23 1.480 1.476 1.479 24 1.517 1.510 1.516

A correlation R₁₁ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁ ²=0.976421, R₁₂ ²=0.972209 and R₁₃ ²=0.971412.

To sum up, in this embodiment, for the first offline drilling fluid, the first prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.035966, b=0.5 and c=0.855298, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In a first embodiment of the on-site measurement, the density of a first on-site drilling fluid is ρ₂=1,003 kg/m³.

The on-site curved pipe is a helical pipe, which has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_(tc2)=0.01051 m:

$\begin{matrix} {d_{{tc}2} = \sqrt{\frac{4^{*}V_{2}}{\pi^{*}{len}_{2}}}} & (11) \end{matrix}$

In this embodiment, a first offline drilling fluid flows through the on-site pipe for i=24 times.

A flow velocity v_(ci) of the first on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_(ci)/ΔL_(ci) are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

Rate Number of flow Temperature of times (lpm) (° C.) ΔP_(ci)/ΔL_(ci) ρ₂ v_(ci) 1 0.8882 32.5 0.590 1003.9 0.1705 2 1.2356 32.5 0.727 1002.7 0.2372 3 1.3858 32.5 0.777 1002.5 0.2660 4 1.4388 32.5 0.830 1003.4 0.2762 5 1.5973 32.5 0.873 1004.1 0.3066 6 1.7200 32.5 0.927 1003.3 0.3302 7 1.8269 32.5 0.979 1004.2 0.3507 8 1.9638 32.5 1.023 1002.7 0.3770 9 2.0876 32.5 1.064 1002.9 0.4007 10 2.3133 32.5 1.169 1002.6 0.4441 11 2.5336 32.5 1.242 1002.8 0.4864 12 2.6463 32.5 1.295 1002.7 0.5080 13 2.8562 32.5 1.350 1002.8 0.5483 14 3.0025 32.5 1.408 1003.2 0.5764 15 3.1087 32.5 1.468 1002.9 0.5968 16 3.2385 32.5 1.525 1003.5 0.6217 17 3.3183 32.5 1.589 1002.9 0.6370 18 3.4960 32.5 1.627 1002.6 0.6711 19 3.6600 32.5 1.694 1003.7 0.7026 20 3.7989 32.5 1.785 1003.3 0.7293 21 4.0419 32.5 1.845 1002.6 0.7759 22 4.0901 32.5 1.904 1002.9 0.7852 23 4.2535 32.5 1.977 1003.3 0.8165 24 4.6207 32.5 2.089 1003.0 0.8870

The first on-site drilling fluid flows through the curved pipe for 24 times, and its friction coefficient f_(ci) in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{matrix} {f_{ci} = {\frac{d_{{tc}2}}{{2\rho_{2}} \star v_{ci}^{2}}*\frac{\Delta P_{ci}}{\Delta L_{ci}}}} & (10) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the first prediction model: ŷ_(1k)=a*D_(nk) ^(b)+c, where the relationship constants between the friction coefficients are respectively: a=0.035966, b=0.5, and c=0.855298.

The Reynolds number R_(ei) of the on-site curved pipe is expressed by formula (12):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}\left( {{a^{*}\left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}^{b} + c} \right)}} & (12) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_(ei) of the on-site curved pipe, a shear stress τw_(i) of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{matrix} {\tau_{wi} = \frac{8{\rho_{2}}^{*}v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$

In this embodiment, an intermediate parameter N_(i) is calculated by a binomial fitting method according to formula (17), which is shown in the table below.

$\begin{matrix} {N_{i} = \frac{d\left( {\ln\;\tau_{w_{i}}} \right)}{d\left( {\ln\frac{8^{*}v_{ci}}{d_{{tc}\; 2}}} \right)}} & (17) \end{matrix}$

A shear rate γ_(i) of the first drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{matrix} {\gamma_{l} = {\frac{8^{*}v_{ci}}{d_{{tc}\; 2}}*\frac{{3^{*}N_{i}} + 1}{4^{*}N_{i}}}} & (16) \end{matrix}$

Number of times f_(ci) R_(ci) τw_(i) N_(i) γ_(i) 1 0.11 158.92 1.47 0.47 166.99 2 0.07 263.11 1.72 0.50 225.78 3 0.06 316.51 1.79 0.51 250.91 4 0.06 320.14 1.91 0.51 259.74 5 0.05 384.57 1.96 0.53 286.06 6 0.05 425.27 2.06 0.53 306.33 7 0.04 459.71 2.15 0.54 323.95 8 0.04 516.12 2.21 0.55 346.44 9 0.04 569.37 2.26 0.55 366.69 10 0.03 649.48 2.44 0.56 403.50 11 0.03 751.29 2.53 0.57 439.27 12 0.03 793.06 2.61 0.58 457.51 13 0.02 908.07 2.66 0.58 491.39 14 0.02 975.29 2.73 0.59 514.94 15 0.02 1008.00 2.84 0.59 532.01 16 0.02 1064.98 2.91 0.60 552.83 17 0.02 1073.93 3.03 0.60 565.62 18 0.02 1186.55 3.05 0.61 594.03 19 0.02 1267.02 3.13 0.61 620.20 20 0.02 1301.98 3.28 0.61 642.33 21 0.02 1458.08 3.31 0.62 680.96 22 0.02 1444.81 3.42 0.62 688.62 23 0.02 1521.06 3.52 0.63 714.54 24 0.01 1750.53 3.61 0.63 772.62

According to the shear stress τw_(i) and the shear rate γ_(i) of the first on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w _(1i) =YP+PV*γ _(i)  (18)

where, PV=0.00354, and YP=0.95267.

a second on-site model:

{circumflex over (τ)}w _(2i) =K*γ _(i) ^(n)  (19),

where K=0.0622, and n=0.6105.

a third on-site model:

{circumflex over (τ)}w _(2i)=τ₀ +K*γ _(i) ^(n)  (20),

where, n=0.7991, K=0.0151, and τ₀=0.581.

Then the following parameters are respectively calculated:

a correlation R₂₁ ² between the shear stress τw_(i) of the first on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂ ² between the shear stress τw_(i) of the first on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃ ² between the shear stress τw_(i) of the first on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{matrix} {R_{21}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w_{i}} - \hat{\tau\; w_{1i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (21) \\ {R_{22}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w_{i}} - \hat{\tau\; w_{2i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (22) \\ {R_{23}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w_{i}} - \hat{\tau\; w_{3i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\;\left( {{\tau\; w} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (23) \end{matrix}$

Through calculation, R₂₁ ²=0.9950, R₂₂ ²=0.9951 and R₂₃ ²=0.9964. The third on-site model has the maximum correlation and is most in line with the actual situation, so the third on-site model is selected as the final model for calculating other viscosity data.

The on-site measurement results of the third on-site model are compared with those of a 6-speed viscometer (Fann35), which shows that the third on-site model is the most suitable.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 PV YP Fann35 (control) 34 7 5 4 3 1 0.5 0.002 1.533 System measurement 34 8.6 5.4 4.3 2.9 1.3 1.2 0.0032 1.1538 device

If the viscosity data calculated by the first on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ6 is 0.9/1=90%, the difference percentage of viscosity corresponding to θ3 is 1.4/0.5=280%, and the difference percentage of YP is 0.5804/1.533=38%. The calculation of the preferred third model of the present invention shows that the difference percentage of viscosity corresponding to θ6 is 0.3/1=30%, the difference percentage of viscosity corresponding to θ3 is 0.7/0.5=140%, and the difference percentage of YP is 0.3792/1.533=25%.

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site model (third on-site model) determined by the correlations of the actual shear stress τw_(i) and the predicted shear stresses of the on-site models are the most accurate.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 PV YP Fann35 (control) 34 7 5 4 3 1 0.5 0.002 1.533 System measurement 34 9.0 5.4 4.2 3.0 1.9 1.9 0.0035 0.9526 device

A second embodiment is described below.

In the second embodiment of offline checking, the offline curved pipe has a total volume V₁=1.04 and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_(ts1)=0.01056 m; and a second offline drilling fluid has a density ρ₁=1,953 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_(tc1)=0.01051 m:

$\begin{matrix} {d_{{tc}\; 1} = \sqrt{\frac{4^{*}V_{1}}{\pi^{*}{len}_{1}}}} & (4) \end{matrix}$

In this embodiment, the second offline drilling fluid flows through the pipe for k=17 times.

A flow velocity v_(sk) of the second offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_(ck) thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_(ck)/ΔL_(ck) of the offline curved pipe and an average pressure difference ΔP_(sk)/ΔL_(sk) of the offline straight pipe are also shown in the table below.

The second offline drilling fluid flows through for 17 times, and its friction coefficient f_(ck) in the offline curved pipe and friction coefficient f_(sk) in the offline straight pipe are calculated by formulas (1) and (2), and are shown in the table below.

$\begin{matrix} {f_{ck} = {\frac{d_{{tc}\; 1}}{2{\rho_{1}}^{*}v_{ck}^{2}}*\frac{\Delta\; P_{ck}}{\Delta\; L}}} & (1) \\ {f_{sk} = {\frac{d_{{ts}\; 1}}{2{\rho_{1}}^{*}v_{sk}^{2}}*\frac{\Delta\; P_{sk}}{\Delta\; L}}} & (2) \end{matrix}$

The corresponding actual friction coefficient ratios y_(k) are shown in the table below, where y_(k)=f_(ck)/f_(sk).

In this embodiment, an average viscosity of the second offline drilling fluid is μ₁=0.02639 Pa·s.

The Dean numbers D_(nk) of the second offline drilling fluid flowing through the curved pipe for 17 times are calculated according to formula (6), which are shown in the table below.

$\begin{matrix} {D_{nk} = {\frac{{\rho_{1}}^{*}{v_{ck}}^{*}d_{{tc}\; 1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}\; 1}}{D_{c1}}}}} & (6) \end{matrix}$

Number ΔP_(ck)/ ΔP_(sk)/ of times ΔL_(ck) ΔL_(sk) v_(sk) v_(ck) f_(sk) f_(ck) y_(k) D_(nk) 1  6.249  8.759 0.80 0.81 0.03 0.04 1.37 119.74 2  7.020  9.999 0.91 0.91 0.02 0.03 1.39 135.57 3  7.447 10.907 0.97 0.98 0.02 0.03 1.43 145.85 4  8.008 12.182 1.07 1.08 0.02 0.03 1.49 166.80 5  8.420 12.888 1.11 1.12 0.02 0.03 1.50 167.88 6  8.583 13.091 1.13 1.14 0.02 0.03 1.49 172.00 7  9.179 14.505 1.21 1.22 0.02 0.03 1.54 184.56 8  9.603 15.250 1.29 1.30 0.02 0.02 1.55 199.90 9 10.128 16.783 1.37 1.38 0.01 0.02 1.62 215.73 10 10.338 17.224 1.41 1.43 0.01 0.02 1.63 224.42 11 10.880 18.257 1.47 1.48 0.01 0.02 1.64 230.18 12 13.825 24.076 1.82 1.83 0.01 0.02 1.70 277.64 13 13.706 24.131 1.83 1.85 0.01 0.02 1.72 284.93 14 14.606 26.401 1.96 1.98 0.01 0.02 1.77 305.94 15 15.228 27.892 2.07 2.09 0.01 0.02 1.79 326.25 16 16.025 29.395 2.14 2.16 0.01 0.02 1.79 333.56 17 16.940 30.332 2.21 2.23 0.01 0.02 1.75 333.69

Three prediction models are fitted by the actual friction coefficient ratios yk and the Dean numbers D_(nk) as follows:

a first prediction model:

ŷ _(1k) =a*D _(nk) ^(b) +c  (3)

where, a=0.0576, b=0.5, and c=0.745.

a second prediction model:

$\begin{matrix} {{\overset{\hat{}}{y}}_{2k} = {1 + \frac{a^{*}D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$

where, a=0.0644, and b=1.4654.

a third prediction model:

ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5)

where, a=0.0231, and b=3.8241.

The predicted friction coefficients of the third prediction models are shown in the table below.

Number of times ŷ_(1k) ŷ_(2k) ŷ_(3k) 1 1.376 1.377 1.379 2 1.416 1.417 1.418 3 1.441 1.442 1.443 4 1.489 1.491 1.490 5 1.492 1.493 1.492 6 1.501 1.502 1.501 7 1.528 1.529 1.528 8 1.560 1.561 1.560 9 1.591 1.593 1.591 10 1.608 1.610 1.608 11 1.619 1.620 1.619 12 1.705 1.705 1.704 13 1.718 1.717 1.717 14 1.753 1.752 1.752 15 1.786 1.784 1.785 16 1.797 1.795 1.796 17 1.798 1.795 1.797

A correlation R₁₁ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁ ²=0.9833, R₁₂ ²=0.9843 and R₁₃ ²=0.9828.

To sum up, in this embodiment, for the second offline drilling fluid, the first prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.0644 and b=1.4654, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In a second embodiment of the on-site measurement, the density of a second on-site drilling fluid is ρ₂=1,300 kg/m³.

The on-site curved pipe has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_(tc2)=0.01051 m:

$\begin{matrix} {d_{{tc}\; 2} = \sqrt{\frac{4^{*}\nabla_{2}}{\pi^{*}{len}_{2}}}} & (11) \end{matrix}$

In this embodiment, a second offline drilling fluid flows through the on-site pipe for i=17 times.

A i-th flow velocity v_(ci) of the second on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_(ci)/ΔL_(ci) are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

Rate Number of flow Temperature of times (lpm) (° C.) ΔPci/ΔLci ρ2 vci 1 4.212 29 8.759 1960.5 0.809 2 4.761 29 9.999 1951.0 0.914 3 5.088 29 10.907 1950.2 0.977 4 5.642 29 12.182 1950.2 1.083 5 5.813 29 12.888 1944.5 1.116 6 5.933 29 13.091 1949.2 1.139 7 6.358 29 14.505 1947.9 1.221 8 6.776 29 15.250 1943.1 1.301 9 7.207 29 16.783 1955.0 1.384 10 7.425 29 17.224 1956.0 1.425 11 7.716 29 18.257 1955.1 1.481 12 9.544 29 24.076 1958.6 1.832 13 9.631 29 24.131 1956.9 1.849 14 10.306 29 26.401 1955.3 1.978 15 10.868 29 27.892 1955.2 2.086 16 11.263 29 29.395 1958.4 2.162 17 11.595 29 30.332 1954.3 2.226

The second on-site drilling fluid flows through the curved pipe for 17 times, and its friction coefficient f_(ci) in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{matrix} {f_{ci} = {\frac{d_{{tc}\; 2}}{2{\rho_{2}}^{*}v_{ci}^{2}}*\frac{\Delta\; P_{ci}}{\Delta\; L_{ci}}}} & (10) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the second prediction model:

${{\overset{\hat{}}{y}}_{2k} = {1 + \frac{a^{*}D_{nk}^{b}}{{70} + D_{nk}}}},$

where the relationship constants between the friction coefficients are respectively: a=0.0644 and b=1.4654.

The Reynolds number R_(ei) of the on-site curved pipe is expressed by formula (12):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}\left( {{a^{*}\left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}^{b} + c} \right)}} & (12) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_(ei) of the on-site curved pipe, a shear stress τw_(i) of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{matrix} {\tau_{wi} = \frac{8{\rho_{2}}^{*}v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$

An intermediate parameter N_(i) is calculated according to formula (17), which is shown in the table below.

$\begin{matrix} {N_{i} = \frac{d\left( {\ln\;\tau_{w_{i}}} \right)}{d\left( {\ln\frac{8^{*}v_{ci}}{d_{{tc}\; 2}}} \right)}} & (17) \end{matrix}$

A shear rate γ_(i) of the second drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{matrix} {\gamma_{i} = {\frac{8^{*}v_{ci}}{d_{tc2}}*\frac{{3^{*}N_{i}} + 1}{4^{*}N_{i}}}} & (16) \end{matrix}$

Number of times f_(ci) R_(ei) τw_(i) N_(i) γ_(i) 1 0.0359 614.0792 16.6964 0.9448 624.1797 2 0.0322 707.0957 18.4418 0.9517 704.3091 3 0.0308 750.4231 19.8311 0.9554 751.8029 4 0.0280 852.8917 21.4602 0.9612 832.4728 5 0.0280 853.1906 22.7020 0.9628 857.2334 6 0.0272 885.4715 22.8450 0.9640 874.7194 7 0.0263 928.1204 25.0111 0.9678 936.3869 8 0.0244 1026.7406 25.6190 0.9714 997.0713 9 0.0236 1074.6661 27.8581 0.9748 1059.5306 10 0.0228 1125.6006 28.2435 0.9765 1091.0656 11 0.0224 1153.9089 29.7386 0.9786 1133.1933 12 0.0193 1418.3999 37.0815 0.9905 1397.4120 13 0.0190 1448.0218 36.9546 0.9910 1409.9395 14 0.0181 1540.8543 39.7384 0.9948 1507.3717 15 0.0172 1654.1138 41.1553 0.9978 1588.2620 16 0.0169 1702.5492 43.0205 0.9998 1645.2677 17 0.0165 1761.7294 43.9646 1.0014 1692.9891

According to the shear stress τw_(i) and the shear rate γ_(i) of the second on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w _(1i) =YP+PV*γ _(i)  (18)

where, PV=0.026, and YP=0.241.

a second on-site model:

{circumflex over (τ)}w _(2i) =K*γ _(i) ^(n)  (19),

where K=0.0278, and n=0.9917.

a third on-site model:

{circumflex over (τ)}w _(2i)=τ₀ +K*γ _(i) ^(n)  (20),

Where, n=0.9991, K=0.0262, and τ₀=0.2146.

Then the following parameters are respectively calculated:

a correlation R₂₁ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{matrix} {R_{21}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{1i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (21) \end{matrix}$ $\begin{matrix} {R_{22}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{2i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (22) \end{matrix}$ $\begin{matrix} {R_{23}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{3i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (23) \end{matrix}$

Through calculation, R₂₁ ²=0.998873, R₂₂ ²=0.996445 and R₂₃ ²=0.998873. The first and third on-site models have the maximum correlation and are most in line with the actual situation, so the first and third on-site models are selected as the final models for calculating other viscosity data.

The on-site measurement results of the first on-site model are compared with those of the 6-speed viscometer, which shows that the first on-site model is the most suitable.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 Fann35 (control) 29 50 27 19 11 1 0.5 System measurement 29 52.42 26.45 17.79 9.13 0.99 0.73 device

The on-site measurement results of the third on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 n K τ0 Fann35 (control) 29 50 27 19 11 1 0.5 0.9015 0.0490 0.26 System measurement 29 52.42 26.44 17.77 9.10 0.94 0.68 0.9991 0.0262 0.21 device

If the viscosity data calculated by the second on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ100 is 2.13/11=19% and the difference percentage of viscosity corresponding to θ6 is 0.46/1=46%. The calculation of the preferred third model of the present invention shows that the difference percentage of viscosity corresponding to θ100 is 1.9/11=17% and the difference percentage of viscosity corresponding to θ6 is 0.06/1=6%. The calculation of the preferred first model of the present invention shows that the difference percentage of viscosity corresponding to θ100 is 1.83/11=17% and the difference percentage of viscosity corresponding to θ6 is 0.01/1=1%.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 Fann35 (control) 29 50 27 19 11 1 0.5 System measurement 29 52.43 26.37 17.64 8.87 0.54 0.27 device

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site models (the first and third on-site models) determined by the correlations of the actual shear stress τw_(i) and the predicted shear stresses of the on-site models are the most accurate.

A third embodiment is described below.

In the third embodiment of offline checking, the offline curved pipe has a total volume V₁=1.04 l and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_(ts1)=0.01056 m; and a third offline drilling fluid has a density ρ₁=1,227 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_(tc1)=0.01051 m:

$\begin{matrix} {d_{{tc}1} = \sqrt{\frac{4^{*}V_{1}}{\pi^{*}{len}_{1}}}} & (4) \end{matrix}$

In this embodiment, the third offline drilling fluid flows through the pipe for k=13 times.

A flow velocity v_(sk) of the third offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_(ck) thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_(ck)/ΔL_(ck) of the offline curved pipe and an average pressure difference ΔP_(sk)/ΔL_(sk) of the offline straight pipe are also shown in the table below.

The third offline drilling fluid flows through the pipe for 13 times, and its friction coefficient f_(ck) in the curved pipe and friction coefficient f_(sk) in the straight pipe are calculated by formulas (1) and (2), and are shown in the table below.

$\begin{matrix} {f_{ck} = {\frac{d_{{tc}1}}{{2\rho_{1}} \star v_{ck}^{2}}*\frac{\Delta P_{ck}}{\Delta L_{ck}}}} & (1) \end{matrix}$ $\begin{matrix} {f_{sk} = {\frac{d_{{ts}1}}{{2\rho_{1}} \star v_{sk}^{2}}*\frac{\Delta P_{sk}}{\Delta L_{sk}}}} & (2) \end{matrix}$

The corresponding actual friction coefficient ratios y_(k) are shown in the table below, where y_(k)=f_(ck)/f_(sk).

In this embodiment, an average viscosity of the third offline drilling fluid is μ₁=0.01455 Pa·s.

The Dean numbers D_(nk) of the third offline drilling fluid flowing through the curved pipe for 13 times are calculated according to formula (6), which are shown in the table below.

$\begin{matrix} {D_{nk} = {\frac{\rho_{1} \star v_{ck} \star d_{{tc}1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}1}}{D_{c1}}}}} & (6) \end{matrix}$

Number ΔP_(ck)/ ΔP_(sk)/ of times ΔL_(ck) ΔL_(sk) v_(sk) _(vck) f_(sk) f_(ck) y_(k) D_(nk) 1 2.634 3.108 0.56 0.56 0.04 0.04 1.15  85.88 2 2.788 3.381 0.61 0.61 0.03 0.04 1.19  96.66 3 2.881 3.555 0.64 0.65 0.03 0.04 1.21 105.05 4 3.050 3.808 0.69 0.70 0.03 0.03 1.22 114.64 5 3.363 4.360 0.79 0.80 0.02 0.03 1.27 134.93 6 3.447 4.622 0.83 0.84 0.02 0.03 1.31 145.22 7 3.555 4.894 0.87 0.88 0.02 0.03 1.35 154.57 8 3.664 5.160 0.91 0.92 0.02 0.03 1.38 166.17 9 3.784 5.453 0.96 0.97 0.02 0.03 1.41 176.43 10 3.909 5.734 1.01 1.02 0.02 0.02 1.43 190.62 11 4.036 6.001 1.05 1.06 0.02 0.02 1.45 198.90 12 4.177 6.287 1.09 1.10 0.02 0.02 1.47 208.59 13 4.215 6.420 1.11 1.12 0.01 0.02 1.49 211.66

Three prediction models are fitted by the actual friction coefficient ratios y_(k) and the Dean numbers D_(nk) as follows:

a first prediction model:

ŷ _(1k) =a*D _(nk) ^(b) +c  (3)

where, a=0.0645, b=0.5, and c=0.5424.

a second prediction model:

$\begin{matrix} {{\overset{\hat{}}{y}}_{2k} = {1 + \frac{a \star D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$

where, a=0.0045, and b=1.9298.

a third prediction model:

ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5)

where, a=0.0025, and b=6.2539.

The predicted friction coefficients of the third prediction models are shown in the table below.

Number of times ŷ_(1k) ŷ_(2k) ŷ_(3k) 1 1.1401 1.1555 1.1547 2 1.1765 1.1827 1.1822 3 1.2035 1.2042 1.2040 4 1.2330 1.2292 1.2291 5 1.2916 1.2828 1.2830 6 1.3196 1.3103 1.3106 7 1.3443 1.3354 1.3358 8 1.3738 1.3668 1.3671 9 1.3991 1.3946 1.3948 10 1.4329 1.4332 1.4333 11 1.4520 1.4557 1.4557 12 1.4739 1.4822 1.4819 13 1.4807 1.4905 1.4902

A correlation R₁₁ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁ ²=0.9923, R₁₂ ²=0.9948 and R₁₃ ²=0.9949.

To sum up, in this embodiment, for the third offline drilling fluid, the third prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.0025 and b=6.2539, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In the third embodiment of the on-site measurement, the density of a third on-site drilling fluid is ρ₃=1,227 kg/m³.

The on-site curved pipe has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_(tc2)=0.01051 m:

$\begin{matrix} {d_{tc2} = \sqrt{\frac{4^{*}V_{2}}{\pi^{*}{le}n_{2}}}} & (11) \end{matrix}$

In this embodiment, the third offline drilling fluid flows through the on-site pipe for i=13 times.

A i-th flow velocity v_(ci) of the second on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_(ci)/ΔL_(ci) are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

Rate Number of flow Temperature of times (lpm) (° C.) ΔP_(ci)/ΔL_(ci) ρ₂ vc_(i) 1 2.926 29 3.11 1227.7 0.5617 2 3.195 29 3.38 1227.5 0.6133 3 3.385 29 3.55 1228.0 0.6497 4 3.640 29 3.81 1226.4 0.6988 5 4.146 29 4.36 1227.3 0.7958 6 4.355 29 4.62 1226.6 0.8360 7 4.563 29 4.89 1226.8 0.8759 8 4.802 29 5.16 1227.3 0.9218 9 5.028 29 5.45 1227.5 0.9652 10 5.313 29 5.73 1226.9 1.0199 11 5.512 29 6.00 1227.9 1.0582 12 5.741 29 6.29 1228.7 1.1021 13 5.815 29 6.42 1226.2 1.1163

The second on-site drilling fluid flows through the curved pipe for 13 times, and its friction coefficient f_(ci) in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{matrix} {f_{ci} = {\frac{d_{tc2}}{{2\rho_{2}} \star v_{ci}^{2}}*\frac{\Delta P_{ci}}{\Delta L_{ci}}}} & (10) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the third prediction model: ŷ_(3k)=1+a*(log₁₀ D_(nk))^(b), where the relationship constants between the friction coefficients are respectively: a=0.0025 and b=6.2539.

The Reynolds number R_(ei) of the on-site curved pipe is expressed by formula (12):

$\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}\left( {1 + {a*\left( {\log_{10}\left( {R_{ei}*\sqrt{\frac{d_{tc2}}{D_{c2}}}} \right)} \right)^{b}}} \right)}} & (12) \end{matrix}$

The Reynolds number R_(ei) of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_(ei) of the on-site curved pipe, a shear stress τw_(i) of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{matrix} {\tau_{wi} = \frac{{8\rho_{2}} \star v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$

An intermediate parameter N_(i) is calculated according to formula (17), which is shown in the table below.

$\begin{matrix} {N_{i} = \frac{d\left( {\ln\tau_{wi}} \right)}{d\left( {\ln\frac{8^{*}v_{ci}}{d_{{tc}2}}} \right)}} & (17) \end{matrix}$

A shear rate γ_(i) of the third drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{matrix} {\gamma_{i} = {\frac{8^{*}v_{ci}}{d_{tc2}}*\frac{{3^{*}N_{i}} + 1}{4^{*}N_{i}}}} & (16) \end{matrix}$

Number of times f_(ci) R_(ei) τw_(i) γ_(i) 1 0.042 438.213 7.072 474.696 2 0.038 491.082 7.521 518.882 3 0.036 534.109 7.764 550.185 4 0.033 589.713 8.124 592.341 5 0.029 699.632 8.888 675.882 6 0.028 739.707 9.271 710.482 7 0.027 779.393 9.661 744.922 8 0.026 838.195 9.953 784.520 9 0.025 885.939 10.327 822.071 10 0.024 970.584 10.518 869.288 11 0.023 1016.091 10.825 902.452 12 0.022 1075.315 11.102 940.432 13 0.022 1080.252 11.316 952.785

According to the shear stress τw_(i) and the shear rate γ_(i) of the third on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w _(1i) =YP+PV*γ _(i)  (18)

where, PV=0.0088, and YP=2.98.

a second on-site model:

{circumflex over (τ)}w _(2i) =K*γ _(i) ^(n)  (19),

where, K=0.6715, and n=0.1126.

a third on-site model:

{circumflex over (τ)}w _(2i)=τ₀ +K*γ _(i) ^(n)  (20),

where, n=0.6716, K=0.1126, and τ₀=0.001.

Then the following parameters are respectively calculated.

a correlation R₂₁ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃ ² between the shear stress τw_(i) of the second on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{matrix} {R_{21}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{1i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (21) \end{matrix}$ $\begin{matrix} {R_{22}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{2i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (22) \end{matrix}$ $\begin{matrix} {R_{23}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - {\tau{\overset{\hat{}}{w}}_{3i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau w}_{i} - \overset{\_}{\tau w}} \right)^{2}}}} & (23) \end{matrix}$

Through calculation, R₂₁ ²=0.996314, R₂₂ ²=0.997775 and R₂₃ ²=0.997775. The second and third on-site models have the maximum correlation and are most in line with the actual situation, so the second and third on-site models are selected as the final models for calculating other viscosity data.

The on-site measurement results of the second on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 n K Fann35 (control) 29 22 14 10 6.5 1 0.8 0.6521 0.1226 System measurement 29 23.1 14.5 11.1 6.9 1.0 0.7 0.6715 0.1126 device

The on-site measurement results of the third on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 n K τ0 Fann35 (control) 29 22 14 10 6.5 1 0.8 0.6835 0.0950 0.4088 System measurement 29 23.1 14.5 11.1 6.9 1.0 0.7 0.6716 0.1126 0.001  device

If the viscosity data calculated by the first on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ100 is 2.2/6.5=35%, the difference percentage of viscosity corresponding to θ6 is 5/1=500%, and the difference percentage of viscosity corresponding to θ3 is 5.1/0.8=639%. The calculation results of the preferred second and third models of the present invention show that the difference percentage of viscosity corresponding to θ100 is 0.4/6.5=7%, the difference percentage of viscosity corresponding to θ6 is 0.1/5=5% and the difference percentage of viscosity corresponding to θ3 is −0.1/0.8=−17%.

Measurement Measuring instrument temperature/° C. θ600 θ300 θ200 θ100 θ6 θ3 Fann35 29 22 14 10 6.5 1 0.8 (control) System 29 23.4 14.6 11.7 8.7 6.0 5.9 measurement device

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site models (the second and third on-site models) determined by the correlations of the actual shear stress τw_(i) and the predicted shear stresses of the on-site models are the most accurate.

The above described are merely preferred embodiments of the present invention, which are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention. 

What is claimed is:
 1. A method for measuring a rheological property of a drilling fluid by using a curved pipe on site, comprising the following steps: step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking; step 2: calculating a Reynolds number R_(ei) of the drilling fluid in an on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and a friction coefficient f_(ci) of the drilling fluid in the on-site curved pipe, wherein i denotes a number of times the drilling fluid flows through the on-site curved pipe, and i is a positive integer not less than 2; step 3: calculating an actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe according to the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe, and calculating a shear rate γ_(i) of the drilling fluid according to the actual shear stress τw_(i); step 4: establishing a plurality of on-site models according to the actual shear stress τw_(i) and the shear rate γ_(i) of the drilling fluid; step 5: determining an optimal on-site model according to correlations between the actual shear stress τw_(i) and predicted shear stresses of the plurality of on-site models; and step 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.
 2. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 1, wherein step 1 comprises: step 11: calculating a friction coefficient f_(ck) of the drilling fluid in an offline curved pipe and a friction coefficient f_(sk) of the drilling fluid in an offline straight pipe, wherein k denotes a number of times the drilling fluid flows through an offline pipe, and k is a positive integer not less than 2; step 12: establishing a plurality of prediction models according to an actual friction coefficient ratio y_(k), wherein, y_(k)=f_(ck)/f_(sk); step 13: determining an optimal prediction model according to correlations between the actual friction coefficient ratio y_(k) and predicted friction coefficient ratios of the plurality of prediction models; and step 14: deriving the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.
 3. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 2, wherein in step 11, f_(ck) is expressed by formula (1): $\begin{matrix} {f_{ck} = {\frac{d_{{tc}1}}{{2\rho_{1}} \star v_{ck}^{2}}*\frac{\Delta P_{ck}}{\Delta L_{ck}}}} & (1) \end{matrix}$ wherein, d_(tc1) denotes an inner diameter of the offline curved pipe, and has a unit of m; ρ₁ denotes a density of an offline drilling fluid, and has a unit of kg/m³; v_(ck) denotes a flow velocity of the drilling fluid at a k-th time in the offline curved pipe, and has a unit of m/s; ΔP_(ck)/ΔL_(ck) denotes a measured average pressure difference in the offline curved pipe, and has a unit of kPa/m; and ΔP_(ck) denotes a total pressure difference in a pipe section with a length of ΔL_(ck), and has a unit of kPa; in step 11, f_(sk) is expressed by formula (2): $\begin{matrix} {f_{sk} = {\frac{d_{{ts}1}}{{2\rho_{1}} \star v_{sk}^{2}}*\frac{\Delta P_{sk}}{\Delta L_{sk}}}} & (2) \end{matrix}$ wherein, d_(ts1) denotes an inner diameter of the offline straight pipe, and has a unit of m; ρ₁ denotes a density of the drilling fluid, and has a unit of kg/m³; v_(sk) denotes a flow velocity of the drilling fluid at the k-th time in the offline straight pipe, and has a unit of m/s; ΔP_(sk)/ΔL_(sk) denotes a measured average pressure difference in the offline straight pipe, and has a unit of kPa/m; and ΔP_(sk) denotes a total pressure difference in a pipe section with a length of ΔL_(sk), and has a unit of kPa.
 4. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 2, wherein the plurality of prediction models at least comprise: a first prediction model: ŷ _(1k) =a*D _(nk) ^(b) +c  (3) a second prediction model: $\begin{matrix} {{\overset{\hat{}}{y}}_{2\; k} = {1 + \frac{a^{*}D_{nk}^{b}}{{70} + D_{nk}}}} & (4) \end{matrix}$ a third prediction model: ŷ _(3k)=1+a*(log₁₀ D _(nk))^(b)  (5) wherein ŷ_(1k) denotes a predicted friction coefficient of the first prediction model; ŷ_(2k) denotes a predicted friction coefficient of the second prediction model; ŷ_(3k) denotes a predicted friction coefficient of the third prediction model; and a, b and c denote the relationship constants between the friction coefficients of the drilling fluid, respectively; wherein, D_(nk) denotes a Dean number of the drilling fluid at a k-th time in the offline curved pipe, and is expressed by formula (6): $\begin{matrix} {D_{nk} = {\frac{\rho_{1}^{*}{v_{ck}}^{*}d_{{tc}\; 1}}{\mu_{1}}*\sqrt{\frac{d_{{tc}\; 1}}{D_{c\; 1}}}}} & (6) \end{matrix}$ wherein μ₁ denotes a viscosity of an offline drilling fluid, and has a unit of Pa·s; and v_(ck) denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s.
 5. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 4, wherein step 13 comprises: expressing a correlation R₁₁ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the first prediction model by formula (7): $\begin{matrix} {R_{11}^{2} = {1 - \frac{\sum\limits_{k = 1}^{m}\left( {y_{k} - \hat{y_{1k}}} \right)^{2}}{\sum\limits_{k = 1}^{m}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (7) \end{matrix}$ expressing a correlation R₁₂ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the second prediction model by formula (8): $\begin{matrix} {R_{12}^{2} = {1 - \frac{\sum\limits_{k = 1}^{m}\left( {y_{k} - \hat{y_{2k}}} \right)^{2}}{\sum\limits_{k = 1}^{m}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (8) \end{matrix}$ expressing a correlation R₁₃ ² between the actual friction coefficient ratio y_(k) and a predicted friction coefficient ratio of the third prediction model by formula (9): $\begin{matrix} {R_{13}^{2} = {1 - \frac{\sum\limits_{k = 1}^{m}\left( {y_{k} - \hat{y_{3k}}} \right)^{2}}{\sum\limits_{k = 1}^{m}\left( {y_{k} - \overset{\_}{y}} \right)^{2}}}} & (9) \end{matrix}$ comparing R₁₁ ², R₁₂ ² and R₁₃ ² in terms of magnitude, and selecting a prediction model with a maximum correlation as an optimal prediction model; wherein m denotes a number of samples; y_(k) denotes the actual friction coefficient ratio; and y denotes an average actual friction coefficient ratio.
 6. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 5, wherein in step 2, f_(ci) is expressed by formula (10): $\begin{matrix} {f_{ci} = {\frac{d_{{tc}\; 2}}{2{\rho_{2}}^{*}v_{ci}^{2}}*\frac{\Delta\; P_{ci}}{\Delta\; L_{ci}}}} & (10) \end{matrix}$ wherein, d_(tc2) denotes an inner diameter of the on-site curved pipe, and has a unit of m; ρ₂ denotes a density of an on-site drilling fluid, and has a unit of kg/m³; v_(ci) denotes a flow velocity of the drilling fluid at an i-th time in the on-site curved pipe, and has a unit of m/s; ΔP_(ci)/ΔL_(ci) denotes a measured average pressure difference in the on-site curved pipe, and has a unit of kPa/m; and ΔP_(ci) denotes a total pressure difference in a pipe section with a length of ΔL_(ci), and has a unit of kPa.
 7. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 6, wherein in step 2, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe is calculated as follows: when an offline model is the first prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (12): $\begin{matrix} {f_{c1} = {\frac{16}{R_{ei}}\left( {{a^{*}\left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}^{b} + c} \right)}} & (12) \end{matrix}$ when the offline model is the second prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (13): $\begin{matrix} {{f_{ci} = {\frac{16}{R_{ei}}*\left\lbrack {1 + \frac{{a^{*}\left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}^{b}}{{70} + \left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)}} \right\rbrack}},} & (13) \end{matrix}$ and when the offline model is the third prediction model, the Reynolds number R_(ei) of the drilling fluid in the on-site curved pipe satisfies formula (14): $\begin{matrix} {f_{ci} = {\frac{16}{R_{ei}}*{\left\lbrack {1 + {a^{*}\left( {\log_{10}\left( {{R_{ei}}^{*}\sqrt{\frac{d_{{tc}\; 2}}{D_{c\; 2}}}} \right)} \right)}^{b}} \right\rbrack.}}} & (14) \end{matrix}$
 8. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 7, wherein in step 3, the actual shear stress τw_(i) of the drilling fluid in the on-site curved pipe is expressed by formula (15): $\begin{matrix} {\tau_{wi} = \frac{8{\rho_{2}}^{*}v_{ci}^{2}}{R_{ei}}} & (15) \end{matrix}$ wherein v_(ci) denotes the flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and ρ₂ denotes the density of the on-site drilling fluid, and has a unit of kg/m³; the shear rate γ_(i) of the drilling fluid is expressed by formula (16): $\begin{matrix} {\gamma_{i} = {\frac{8^{*}v_{ci}}{d_{{tc}\; 2}}*\frac{{3^{*}N_{i}} + 1}{4^{*}N_{i}}}} & (16) \end{matrix}$ wherein, N is expressed by formula (17): $\begin{matrix} {N_{i} = {\frac{d\left( {\ln\tau_{w_{i}}} \right)}{d\left( {\ln\frac{8^{*}v_{ci}}{d_{{tc}\; 2}}} \right)}.}} & (17) \end{matrix}$
 9. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 8, wherein the plurality of on-site models at least comprise: a first on-site model: {circumflex over (τ)}w _(1i) =YP+PV*γ _(i)  (18) a second on-site model: {circumflex over (τ)}w _(2i) =K*γ _(i) ^(n)  (19), a third on-site model: {circumflex over (τ)}w _(2i)=τ₀ +K*γ _(i) ^(n)  (20), wherein YP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa; PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s; n denotes a fluidity index of the on-site drilling fluid, and is dimensionless; K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; and τ₀ denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.
 10. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 9, wherein step 5 comprises: expressing a correlation R₂₁ ² between the actual shear stress τw_(i) and a predicted shear stress of the first on-site model by formula (21): $\begin{matrix} {R_{21}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \hat{\tau\; w_{1\; i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (21) \end{matrix}$ expressing a correlation R₂₂ ² between the actual shear stress τw_(i) and a predicted shear stress of the second on-site model by formula (22): $\begin{matrix} {R_{22}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \hat{\tau\; w_{2\; i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (22) \end{matrix}$ expressing a correlation R₂₃ ² between the actual shear stress τw_(i) and a predicted shear stress of the third on-site model by formula (23): $\begin{matrix} {R_{23}^{2} = {1 - \frac{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \hat{\tau\; w_{3\; i}}} \right)^{2}}{\sum\limits_{i = 1}^{m}\left( {{\tau\; w_{i}} - \overset{\_}{\tau\; w}} \right)^{2}}}} & (23) \end{matrix}$ comparing R₂₁ ², R₂₂ ² and R₂₃ ² in terms of magnitude, and selecting an on-site model with a maximum correlation as a final model; wherein m denotes a number of samples; τw_(i) denotes the actual shear stress; and τw denotes an average actual shear stress. 